In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time (so long as the strength of the perturbation is independent of time) and is proportional to the strength of the coupling between the initial and final states of the system (described by the square of the matrix element of the perturbation) as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.
Although the rule is named after Enrico Fermi, most of the work leading to it is due to Paul Dirac, who twenty years earlier had formulated a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.^{[1]}^{[2]} It was given this name because, on account of its importance, Fermi called it "golden rule No. 2".^{[3]}
Most uses of the term Fermi's golden rule are referring to "golden rule No. 2", but Fermi's "golden rule No. 1" is of a similar form and considers the probability of indirect transitions per unit time.^{[4]}
Fermi's golden rule describes a system that begins in an eigenstate of an unperturbed Hamiltonian H_{0} and considers the effect of a perturbing Hamiltonian H' applied to the system. If H' is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If H' is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency ω, the transition is into states with energies that differ by ħω from the energy of the initial state.
In both cases, the transition probability per unit of time from the initial state to a set of final states is essentially constant. It is given, to first-order approximation, by
The standard way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition.^{[5]}^{[6]}
Derivation in time-dependent perturbation theory | |
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Statement of the problem EditThe golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H' of the Hamiltonian. The total Hamiltonian is the sum of an “original” Hamiltonian H_{0} and a perturbation: . In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system , with . Discrete spectrum of final states EditWe first consider the case where the final states are discrete. The expansion of a state in the perturbed system at a time t is . The coefficients a_{n}(t) are yet unknown functions of time yielding the probability amplitudes in the Dirac picture. This state obeys the time-dependent Schrödinger equation: Expanding the Hamiltonian and the state, we see that, to first order, This equation can be rewritten as a system of differential equations specifying the time evolution of the coefficients : For a weak constant perturbation H' that turns on at t = 0, we can use perturbation theory. Namely, if , it is evident that , which simply says that the system stays in the initial state . For states , becomes non-zero due to , and these are assumed to be small due to the weak perturbation. The coefficient which is unity in the unperturbed state, will have a weak contribution from . Hence, one can plug in the zeroth-order form into the above equation to get the first correction for the amplitudes : The probability of transition from the initial state (ith) to the final state (fth) is given by It is important to study a periodic perturbation with a given frequency since arbitrary perturbations can be constructed from periodic perturbations of different frequencies. Since must be Hermitian, we must assume , where is a time independent operator. The solution for this case is^{[7]} Consider now the case where the perturbation frequency is such that where is a small quantity. Unlike the previous case, not all terms in the sum over in the above exact equation for matters, but depends only on and vice versa. Thus, omitting all other terms, we can write The two independent solutions are and the constants and are fixed by the normalization condition. If the system at is in the state, then the probability of finding the system in the state is given by which varies periodically between and , that is to say, the system periodically switches from one state to the other. The situation is different if the final states are in the continuous spectrum. Continuous spectrum of final states EditSince the continuous spectrum lies above the discrete spectrum, and it is clear from the previous section, major role is played by the energies lying near the resonance energy , i.e., when . In this case, it is sufficient to keep only the first term for . Assuming that perturbations are turned on from time , we have then Therefore, the transition probability per unit time, for large t, is given by Note that the delta function in the expression above arises due to the following argument. Defining the time derivative of is , which behaves like a delta function at large t (more more information, please see Sinc function#Relationship to the Dirac delta distribution). The constant decay rate of the golden rule follows.^{[8]} As a constant, it underlies the exponential particle decay laws of radioactivity. (For excessively long times, however, the secular growth of the a_{k}(t) terms invalidates lowest-order perturbation theory, which requires a_{k} ≪ a_{i}.) |
Only the magnitude of the matrix element enters the Fermi's golden rule. The phase of this matrix element, however, contains separate information about the transition process. It appears in expressions that complement the golden rule in the semiclassical Boltzmann equation approach to electron transport.^{[9]}
While the Golden rule is commonly stated and derived in the terms above, the final state (continuum) wave function is often rather vaguely described, and not normalized correctly (and the normalisation is used in the derivation). The problem is that in order to produce a continuum there can be no spatial confinement (which would necessarily discretise the spectrum), and therefore the continuum wave functions must have infinite extent, and in turn this means that the normalisation is infinite, not unity. If the interactions depend on the energy of the continuum state, but not any other quantum numbers, it is usual to normalise continuum wave-functions with energy labelled , by writing where is the Dirac delta function, and effectively a factor of the square-root of the density of states is included into .^{[10]} In this case, the continuum wave function has dimensions of , and the Golden Rule is now
The following paraphrases the treatment of Cohen-Tannoudji.^{[10]} As before, the total Hamiltonian is the sum of an “original” Hamiltonian H_{0} and a perturbation: . We can still expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system, but these now consist of discrete states and continuum states. We assume that the interactions depend on the energy of the continuum state, but not any other quantum numbers. The expansion in the relevant states in the Dirac picture is
Substituting into the time-dependent Schrödinger equation
The Fermi golden rule can be used for calculating the transition probability rate for an electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor, and also for when the electron recombines with the hole and emits a photon.^{[12]} Consider a photon of frequency and wavevector , where the light dispersion relation is and is the index of refraction.
Using the Coulomb gauge where and , the vector potential of the EM wave is given by where the resulting electric field is
For a charged particle in the valence band, the Hamiltonian is
From here on we have transition probability based on time-dependent perturbation theory that
For the initial and final states in valence and conduction bands respectively, we have and , and if the operator does not act on the spin, the electron stays in the same spin state and hence we can write the wavefunctions as Bloch waves so
Finally, we want to know the total transition rate . Hence we need to sum over all initial and final states (i.e. an integral of the Brillouin zone in the k-space), and take into account spin degeneracy, which through some mathematics results in
where is the joint valence-conduction density of states (i.e. the density of pair of states; one occupied valence state, one empty conduction state). In 3D, this is
Finally we note that in a general way we can express the Fermi golden rule for semiconductors as^{[13]}
In a scanning tunneling microscope, the Fermi golden rule is used in deriving the tunneling current. It takes the form
When considering energy level transitions between two discrete states, Fermi's golden rule is written as
Fermi's golden rule predicts that the probability that an excited state will decay depends on the density of states. This can be seen experimentally by measuring the decay rate of a dipole near a mirror: as the presence of the mirror creates regions of higher and lower density of states, the measured decay rate depends on the distance between the mirror and the dipole.^{[15]}^{[16]}